LASCAD Tutorial No. 4: Dynamic analysis of multimode competition and Q-Switched operation

LASCAD Tutorial No. 4: Dynamic analysis of multimode competition and Q-Switched operation Revised: January 17, 2014 Copyright 2014 LAS-CAD GmbH

Table of Contents 1 Table of Contents 1 Introduction... 3 2 Modeling Laser power output, mode competition, beam quality for cw operation... 4 2.1 Tab Gaussian Modes... 4 2.2 Tab Rate Equations... 4 2.3 Tab CW Operation... 5 2.4 Beam Quality... 5 2.5 Results... 5 3 Modeling of Q-Switch Operation... 8 3.1 CW Pumping... 8 3.2 Results for CW Pumping... 9 3.3 Pulsed Pumping... 10 4 Modeling the Effect of Apertures... 12 4.1 Hard-Edged Apertures and Mirrors... 12 4.2 Super-gaussian Ouput Mirrors... 14 5 Conclusions... 16 Note: The numerical results shown in this tutorial, such as in Tab. 1, 2 and 3 are obtained, if for the computation the full version of the LASCAD program is used. If the demo version is used the results are slightly different, since the wavelength is fixed to 1.134 nm which differs from the correct wavelength of a Nd:YAG laser.

Tab Gaussian Modes 3 1 Introduction The purpose of the Dynamic Multimode Analysis (DMA) is to analyze multimode and Q-switch operation of lasers. To this end the transverse mode structure in the cavity is approximated by a set of M Hermite-Gaussian (HG) or Laguerre-Gaussian (LG) modes. Since HG and LG modes represent sets of orthogonal eigenfunctions with different eigenfrequencies, it is assumed that each transverse mode oscillates independently, and therefore the influence of short-time interferences between the modes can be neglected on average. Based on this assumption, the interaction between the population inversion density and the photon numbers in the oscillating modes is described by the following set of time dependent 3D rate equations S C M ( t) = Si( t) i=1,,m, (1) i= 1 S t i = cσ n A Ω A NS s dv i i Si τ C, (2) N t cσ = N S n A C s C N τ f + R p N dop N N dop (3) Eqs.(1 to 3) describe the interaction between the population inversion density N(x,y,z,t), the total photon number S C (t) in the cavity with associated normalized photon density distribution s C (x,y,z,t), and the photon numbers S i (t) belonging to the individual transverse modes with associated normalized photon density distributions s i (x,y,z,t) of the individual modes. The obtained time dependent photon numbers S i (t) in the individual modes and the photon density distributions s i (x,y,z,t) are used to describe their contribution to the whole transverse mode structure. In this way, a time dependent analysis of mode competition effects and multimode behavior of the cavity is obtained. This result is used to compute beam quality and laser power output, and to describe time dependence of Q-switch operation. In Eqs.(2 and 3) the following parameters are used: refractive index of the active medium, n A c vacuum speed of light, N(x,y,z,t) = N 2 N 1 population inversion density (N 1 ~ 0), R η P P = η P / hν pump rate, P P a (x,y,z) σ a p pump efficiency, absorbed pump power density, effective cross section of stimulated emission, τ C mean life time of laser photons in the cavity, τ f spontaneous fluorescence life time of upper laser level, N dop doping density.

4 Modeling Laser power output, mode competition, and beam quality for cw operation Eqs. (2 and 3) refer to a 4-level-system. Multimode analysis of quasi 3-level systems is under development. Fast decay rates between level 3 (pump level) and 2 (upper laser level), and between level 1 (lower laser level) and 0 are assumed. Mathematical relations between the above parameters and the methods used to compute laser power output, and to model Q-switch operation and effects of apertures are described in detail in the manual, or can be shown by clicking the menu item "Help DMA Code" of the DMA activation window of LASCAD. The following is a practical guide to the DMA Code. It explains how to define input values for the parameters shown in the individual tabs of the DMA GUI appropriately to model cw multimode operation, Q-switched operation, and the effect of apertures. 2 Modeling Laser power output, mode competition, and beam quality for cw operation To work with the DMA code it is necessary to insert a thermally lensing crystal into the cavity. For instance follow the instructions in Tutorial No. 1 to prepare a model for an end pumped crystal. Alternativaly, you can directly activate the cavity configuration according to Tutorial No. 1 by opening the project file Tutorial_1.lcd, which can found in the subdirectory "Tutorials" of the LASCAD application directory. After you have executed the FEA code, and have inserted the crystal in the mode plot window, select Dynamic Multimode Analysis in the main LASCAD menu to open the DMA activation window. In this window click the button Open GUI for DMA to open the window Dynamic Multimode Analysis. This window has 5 tabs whose entries are described as follows. 2.1 Tab Gaussian Modes Open this tab to select the Type of gaussian modes being used to approximate the laser mode structure. Select Hermite-Gaussian modes if the mode structure is astigmatic. For rotational symmetric cavity configurations it may be preferable to select Laguerre-Gaussian modes. However, if a high number of transverse modes has to be taken into account it is recommended to use Hermite-Gaussian modes. High order Laguerre-Gaussian modes are more complicated to handle numerically. Selection of mode type also can be carried through automatically by the program based on the astigmatism of the laser mode structure. Maximum transverse mode order defines the highest transverse mode order N max taken into account in x- as well as in y-direction. However, since the total number M of modes increases according to N 2 max+ 2N max + 1 the total number M can become very large with increasing N max so the computation takes a long time. If a large N max is defined, it is necessary to make the Number of grid points in x-and y- direction large enough to resolve the oscillations of the transverse intensity distribution of the higher order transverse modes. Otherwise, this number can have the same size as the number of transverse grid points used for thermal FEA. The latter one also holds Number of grid points in z-direction. Definition of the Stretch factor in x- and y-direction related to beam diameter strongly depends on N max and on the pump light distribution as described in the online help. For N max = 0 it may be necessary to define a stretch factor equal to 2 to take into account the overlap between fundamental mode and absorbed pump power distribution. 2.2 Tab Rate Equations Though some entries of this tab already can be defined in the window Laser Power Output of the LASCAD program, it is necessary to define them in this tab once more. For a comparison

Tab CW Operation 5 with the results obtained for time independent laser power output, the entries must be defined identical. 2.3 Tab CW Operation The default value of 10 ns for Time resolution turned out to be appropriate for common laser configurations. However, this entry, as well as the length of the Time period used for simulation, should be controlled by inspecting the results of the calculation. 2.4 Beam Quality The beam quality factors M x ² and M y ² are computed according to Siegman and Townsend 1 using the expressions and M 2 x M ( 2pi + 1) ( t) = c ( t) (4) i= 1 i M 2 y M ( 2qi + 1) ( t) = c ( t). (5) i= 1 Here p i and q i are the transverse mode orders of the i-th gaussian mode in x- and y-direction, respectively. The coefficients c i (t) are defined by c ( t) P P i, out C, out ( t) ( t) i i =, (6) where P C,out (t) is the total power output at time t, and P i,out (t) are the power outputs of the individual modes. 2.5 Results After clicking Calculate in the DMA GUI, the window DMA Calculation opens, showing the progress of the computation. Among the information shown in this window, for instance, the value for Power output for TEM00 mode using CW time independent recursion formula can be used for comparison with the result obtained by the corresponding computation, which can be started by the use of the LASCAD window Laser Power Output. The power output of individual modes averaged over the last quarter of simulation time is shown at the end of the calculation. The time average is restricted to the last quarter of simulation time, to reduce the influence of output spiking, as shown in Fig. 1. If spiking extends into this last quarter, it is recommended to increase the entry for the Time period of simulation in the tab CW Operation. In the following, if nothing else is specified, physical quantities are assumed to be averaged in the above way. If, for the calculation, the cavity configuration according to Tutorial No. 1 is used together with the default settings for entries of the tabs of the DMA GUI except of N max = 3, the results shown in Tab. 1 are obtained. 1 A.E. Siegman and S. Townsend, "Output beam propagation and beam quality from a multimode stable-cavity laser", IEEE Journal Quantum Electron., 29, 1212 1217 (1993)

6 Modeling Laser power output, mode competition, and beam quality for cw operation Mode (0,3): 1.23341 Mode (0,2): 0.598921 Mode (0,1): 0.560508 Mode (1,2): 0.550806 Mode (3,0): 0.545464 Mode (1,3): 0.491664 Mode (0,0): 0.456887 Mode (3,1): 0.417439 Mode (1,1): 0.413295 Mode (1,0): 0.346157 Mode (3,3): 0.323292 Mode (2,2): 0.321322 Mode (2,1): 0.305183 Mode (2,0): 0.160791 Mode (3,2): 0.149285 Mode (2,3): 7.35643e-040 Tab. 1: Power output [W] of individual modes averaged over last quarter of simulation time obtained for the cavity configuration according to Tutorial No. 1 These results clearly show the influence of the strongly astigmatic distribution of the absorbed pump power density used in Tutorial No. 1, which has a gaussian distribution in the x-z-plane and a super-gaussian distribution with exponent 10 in the y-z-plane. Accordingly, the power of Modes (n,m) and (m,n) is not equal, for instance, Mode (0.3) has a power of 1.31 W, and the Mode (3,0) only 0.57 W. Correspondingly, the obtained beam qualities M x ² = 3.24 and M y ² = 4.22 are different for x- and y-direction. The total power output is 6.87 W. After closing DMA and LASCAD, the numerical results also can be shown by opening the file output.txt in the corresponding DMA directory. To visualize results, open the DMA Viewer with the button Show Results of the DMA GUI. The drop-down box in the lower part of the Viewer window allows for showing several important 2D and 3D plots. An important plot is the power ouput over time, which is shown in figure 1. The computation starts with population inversion density N(x,y,z,t)=0. Since this is different from equilibrium condition, a spiking behavior can be seen at the beginning, which attenuates with increasing time, and finally approaches a constant value. Dependent on the cavity configuration, it may be necessary to increase the Time period used for simulation to achieve convergence.

Results 7 Fig. 1. Power output over time Power output over time also can be shown for individual modes separately. Also interesting are 2D plots of the beam quality over time and the spot of the Mode (0,0) along the cavity axis. An interesting 3D plot is the Beam profile in the crystal, showing the superposition of the individual transverse modes, according to their contribution to the total power output. An example, once more based on the cavity configuration of Tutorial No.1, is shown in Fig. 2. This figure again shows the influence of the strongly astigmatic distribution of the absorbed pump power density used in Tutorial No. 1. The intensity distribution is strongly astigmatic. Transverse modes with higher order in y-direction are dominant.

8 Modeling of Q-Switched Operation Fig. 2. Beam profile in the crystal 3 Modeling of Q-Switched Operation Two different modes of pumping are available, cw pumping and pulsed pumping which can be selected from a drop-down box in the frame Pumping in the Tab Q-switch. Pulsed pumping can be used to create an approximate model for passive Q-switching, see section 3.3. 3.1 CW Pumping For cw pumping a predefined number of subsequent pulses can be computed, which are triggered with constant repetition frequency. During the load period onset of laser oscillation is suppressed by introducing a high artificial cavity loss in the rate equations, which can be defined in the box Q-switch induced loss during load phase. The default value 0.8 of this parameter is usually appropriate. Since no stimulated emission takes place during the load period, a high population inversion is generated. If an opening period >0 is defined, the artificial Q-switch loss is not removed at once, but continuously reduced to the normal cavity loss during the defined opening period. However, this parameter only has a minor influence on pulse energy and shape. Appropriate definition of the pulse period is very important. This quantity does not represent the physical pulse width, it only defines a time domain used for the computation of the pulse. Since population inversion and photon density change strongly during pulse development, it is necessary to define a large number of time steps during the pulse period to get short time steps. Since pulse build-up can be delayed dependent on the cavity configuration, it may be necessary to

Results for CW Pumping 9 make the pulse period much longer than the pulse width to prevent the pulse extending into the relaxation period. Load period + opening period + pulse period must be smaller than the pulse repetition period. The remaining time is the relaxation period, representing a buffer zone between pulse period and new load period, if multiple pulses are computed. The number of time steps during the relaxation period can be small, since population inversion and photon density do not change much. 3.2 Results for CW Pumping If the cavity configuration according to Tutorial No. 1 is used once more together with the default settings for entries of the tabs of the DMA GUI, except that only the fundamental mode is involved (N max = 0), the calculation delivers the following result for the last pulse from a series of three: Power output averaged over pulse repetition period [W] = 2.92419 Pulse energy [mj] = 0.194946 Pulse width (FWHM) [ns] = 5.15 Average Beam Quality M^2 in x-direction = 1 Average Beam Quality M^2 in y-direction = 1 Peak power output 32216.1 [W] at time 0.000198894 [s] The peak power output provides important information concerning damaging of optical components. After zooming in the 2D plot Power output over time, as described in DMA Viewer Help, the shape of the pulses can be visualized. An example is displayed in Fig. 3, showing a typical pulse shape. Fig. 3. Typical pulse shape for cw pumping

10 Modeling of Q-Switched Operation 3.3 Pulsed Pumping Alternatively to cw pumping, Pulsed synchronous with Q-switch can be selected as Mode of pumping. This input can be used to create an approximate model for passive Q-switching. In case of pulsed pumping it is assumed that pumping starts and ends synchronous with the load period. Since the frequency of pulsed pumping is typically much smaller than the pulse repetition frequency in case of cw pumping, the time-lag between two pulses is large. Therefore, we have assumed, that no interaction between subsequent pulses takes place, and have restricted the number of pulses to one. The time period used for computation ends with the pulse period. Nevertheless the Pulse repetition frequency has to be defined by the following reason. The present version of LASCAD uses a time independent code for the thermal analysis. Therefore, in case of pulsed pumping, the time averaged pump power has to be used as input for the FEA. The accuracy of this approximation is sufficient, if the pump frequency is greater than 60 Hz. To obtain the real pump power during the pump pulse, LASCAD divides the value of the pump power defined in the FEA window by the product Pulse repetition frequency x Load period. Therefore both entries have to be defined in the tab Q-switch. For low pump frequencies, or single shot pumping, a time dependent thermal FEA is under development. Since for pulsed pumping a higher population inversion can usually be achieved, it is preferable to use a smaller value for the reflectivity of the output mirror than for cw pumping. Using again the configuration according to Tutorial No. 1 with N max = 0 and Reflectivity of output mirror = 0.5 in the tab Rate Equations, together with the entries in tab Q-switch shown in Fig. 4, the following results are obtained: Pulse 0: M^2_x = 1 M^2y =1 Power output averaged over pulse repetition period [W] = 71.8257 Pulse energy [mj] = 14.3687 Pulse width (FWHM) [ns] = 0.495 Average Beam Quality M^2 in x-direction = 1 Average Beam Quality M^2 in y-direction = 1 Peak power output 2.3572e+007 [W] at time 0.000200002 [s] Power averaged over whole simulation time [W]: 71.8257, with whole simulation time [s]: 0.00020005. Power output of individual modes averaged over computation period [W]: Mode (0,0): 71.8257 Power output summed up over all considered modes = 71.8257

Pulsed Pumping 11 Fig. 4. Entries of tab Q-switch for pulsed pumping Using the input values shown in Fig. 4 the pulse shape shown in Fig. 5 is obtained.

12 Modeling the Effect of Apertures Fig. 5. Pulse shape obtained for pulsed pumping 4 Modeling the Effect of Apertures The entries of the tab Apertures allow for modeling hard-edged and gaussian apertures, as well as hard edged, gaussian and super-gaussian mirrors. 4.1 Hard-Edged Apertures and Mirrors The shape of hard-edged apertures or mirrors can be elliptical or rectangular. Select the corresponding item in the drop-down box Type of Aperture. If, for the position of the aperture, a value < 0 is defined, the program assumes that the aperture is represented by the output mirror (right mirror). The radii of the aperture can be different in x- and y-direction. Use again the configuration according to Tutorial No. 1 with N max = 3 as in Sect. 2.4 together with a circular end mirror with radius 0.2 mm, and run the calculation for cw operation. The obtained results shown in Tab. 2 indicate that, different from Tab. 1, due to the finite radius of the output mirror, only the modes (0,1), (1,0), and (0,0) contribute to the output power. This result is confirmed, if we are looking at the beam profile in the crystal shown in Fig. 6, and compare it with Fig. 2. Correspondingly, the power output is reduced from 6.9 to 4 W, compared with Sect. 2.5 where no aperture is applied. However, the beam qualities are improved to M x ² = 1.71 and M y ² = 1.92.

Hard-Edged Apertures and Mirrors 13 Mode (0,1): 1.83459 Mode (1,0): 1.42093 Mode (0,0): 0.721171 Mode (1,1): 5.85802e-042 Mode (2,0): 4.07249e-042 Mode (0,2): 2.36935e-042 Mode (2,1): 9.62273e-043 Mode (1,2): 7.98643e-043 Mode (3,0): 6.69612e-043 Mode (0,3): 4.63281e-043 Mode (2,2): 3.62786e-043 Mode (3,1): 3.49377e-043 Mode (1,3): 2.9093e-043 Mode (3,2): 2.12117e-043 Mode (2,3): 2.08619e-043 Mode (3,3): 1.53504e-043 Tab. 2. Power output of individual modes averaged over last quarter of simulation time. For the computation the cavity configuration according to Tutorial No. 1 has been used, but with finite radius of the output mirror equal to 0.2 mm Fig. 6. Beam profile in the crystal for a cavity with output mirror of radius 0.2 mm

14 Modeling the Effect of Apertures 4.2 Super-gaussian Ouput Mirrors The tab Apertures allows for defining gaussian and super-gaussian output mirrors. For this purpose, the option Gaussian has to be selected for Type of aperture and a value <0 has to be entered for the position of the aperture. In this case, the reflectivity of the output mirror is defined by SG SG x y R( x, y) R = R0 exp 2 2 + min wtrx w (4) try using a super-gaussian exponent SG >= 2. R min is a peripheral bottom reflectivity. For R 0 the program uses the value R 0 = R out - R min where R out has to be defined as Reflectivity of output mirror in the tab Rate Equations. Since super-gaussian apertures cannot be taken into account in the ABCD matrix algorithm, the inner part of the super-gaussian distribution is fitted by a gaussian approximation dependent on SG, as shown in Fig. 7 for SG = 4. This approximation seems to be appropriate, since only modes with a radius < w trx (y) deliver a significant contribution to overall mode shape. The obtained gaussian fit is used in the ABCD matrix algorithm. Fig. 7. Gaussian fit of a super gaussian reflectivity distribution with SG=4 Once more, we use the configuration according to Tutorial No. 1 with N max = 3, as in Sect. 2.4 and 4.1, together with the inputs shown in Fig. 8. Running the calculation for cw operation, we obtain the results shown in Tab. 3. Now, as one can see, due to the super-gaussian output mirror, only the fundamental mode (0,0) is excited. Correspondingly, the beam profile displayed in Fig. 9 shows the well known shape of a gaussian fundamental mode. Surprisingly, the power of this mode is 4.22 W. This is relatively high compared with the calculation presented in Sect. 2.4, which delivered 6.87 W for the total power of all modes. Compared with the result for a hard-edged aperture the power output increases from 4 to 4.22 W for a supergaussian aperture, whereas the beam quality improves considerably to 1 for x and y- direction.

Super-gaussian Ouput Mirrors 15 Fig. 8. Entries of tab Q-switch Apertures for an output mirror with super-gaussian reflectivity Mode (0,0): 4.22301 Mode (0,1): 3.01251e-041 Mode (1,0): 1.1211e-041 Mode (0,2): 8.25512e-042 Mode (1,1): 6.72075e-042 Mode (0,3): 4.86947e-042 Mode (1,2): 4.73229e-042 Mode (2,0): 4.64064e-042 Mode (2,1): 4.03505e-042 Mode (1,3): 3.73545e-042 Mode (2,2): 3.60176e-042 Mode (2,3): 3.24972e-042 Mode (3,0): 3.24888e-042 Mode (3,1): 3.21808e-042 Mode (3,2): 3.16512e-042 Mode (3,3): 3.05088e-042 Tab. 3. Power output [W] of individual modes averaged over last quarter of simulation time. For the computation the cavity configuration according to Tutorial No. 1 has been used, but with a super-gaussian output mirror, according to the entries shown in Fig. 8.

16 Conclusions Fig. 9. Beam profile in the crystal for a cavity with super-gaussian output mirror 5 Conclusions The results presented in the above sections demonstrate, that the DMA code for dynamic analysis of multimode competition and Q-switch operation is a very versatile tool to compute the time dependent interaction of a set of individual transverse modes, beam quality, laser power output, intensity distribution of the transverse mode structure, pulse shape for cw and pulsed pumping, peak intensity of pulses etc.. In this way it provides the laser engineer with important information to optimize a laser cavity configuration.